Finance Questions - Value at Risk and Cost of Capital

1. Value at Risk (VaR)
Consider a managed portfolio which has expected return of 20% per annum and return standard deviation of 60% per annum, in an economy in which the annual risk-free rate is 8%, the expected return on the market portfolio is 13% per annum, and the return standard deviation of the market portfolio is 20% per annum. :
(a) Determine the beta and return correlation of the managed portfolio with the market portfolio, if CAPM holds.
(b) Determine the 1-year, 1% VaR of a $1 million investment in the managed portfolio, and the 1-year, 1% VaR of a $1 million investment in the market index, if the market value of managed portfolio has the dynamics of Geometric Brownian Motion.
(c) Determine the 2-year, 5%, VaR of a $1 million investment in the managed portfolio, and the 2-year, 5% VaR of a $1 million investment in the market index, if the market value of managed portfolio has the dynamics of Geometric Brownian Motion.

2. Cost of Capital and Hidden Balance Sheet Effects
A firm has $100 million of long-term (perpetual) debt with a 5% coupon rate on par value, $50 million market value of equity, and $50 million of pension assets. The firm’s equity beta is 1.5, the expected return on the market portfolio is 15% per year, the standard deviation of the market portfolio is 20% per year, and the risk-free rate is 5% per year and stays constant through time. Assume the pension fund is fully funded and the cost of corporate debt is the risk-free rate. Treat corporate tax rate = 0%.
(a) What would be your estimate of the cost of capital for the firm’s operating assets, considering only the on-balance sheet assets and liabilities (neglecting the pension plan)?
(b) Considering the economic balance sheet (including the pension plan), determine the cost of capital for the firm’s operating assets given under these scenarios.
i) The pension plan is fully invested in long-term risk-free bonds.
ii) The pension plan is fully invested in the equity market-portfolio index.
iii) The pension plan is fully invested in long-term risk-free bonds, and enters into a swap to receive the total market index return and pay the total long-term risk-free bond return on a notional amount of $50 million.

3. VaR of the Firm’s Assets as an Approach to Risk Accounting
Consider the firm in the preceding problem. Suppose that its pension plan assets are fully invested in the equity market-portfolio index (case ii) and that the correlation between the return on its operating assets and the market portfolio return is 0.50 [market standard deviation = 20%].
(a) Determine the 1-year 1% VaR of the firm’s total assets (operating and pension), if the market value of the total assets follows Geometric Brownian Motion.
(b) What would be the 1-year 1% VaR of the firm’s total assets if the firm replaced all its equity pension assets with long-term risk-free bonds, if the market value of the total assets follows a Geometric Brownian Motion?
(c) From your answers to (a) and (b), what percentage of the firm’s total risk capital is being allocated to non-operating-asset risk of the pension fund assets?

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